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a/advanced-math/exercise/2-function/function.tex b/advanced-math/exercise/2-function/function.tex index 4de6ea3..6e7de98 100644 --- a/advanced-math/exercise/2-function/function.tex +++ b/advanced-math/exercise/2-function/function.tex @@ -166,9 +166,17 @@ $\therefore a=1,b=e$。 \subsection{罗尔定理} -罗尔定理在判断不等式时一般用于零点的状况。 +\subsubsection{寻找原函数} -\subsubsection{直接式子} +通过乘积求导公式$(uv)'=u'v+uv'$的逆运算来构造辅助函数。 + +如$f(x)f'(x)$,作$F(x)=f^2(x)$,$[f'(x)]^2+f(x)f''(x)$,作$F(x)=f(x)f'(x)$,$f'(x)+f(x)\varphi'(x)$,作$F(x)=f(x)e^{\varphi(x)}$。 + +即证明什么就构造他的原函数为函数式子。 + +\subsubsection{零点情况} + +\paragraph{直接式子} \leavevmode \medskip 需要证明所给式子的导数是否在该区间为0即可。 @@ -180,7 +188,7 @@ $\therefore a=1,b=e$。 由罗尔定理得知$\exists\xi\in(x_1,x_2)\subset(0,1)$,使得$f'(\xi)=0$,但是$f'(x)=3x^2-3$在$(0,1)$上不超过0,所以$\xi$不存在,从而多项式$f(x)=x^3-3x+a$在$[0,1]$上不可能有两个零点。 -\subsubsection{含参数式子} +\paragraph{含参数式子} \leavevmode \medskip 若所求式子是一个含参数,那么其一定还有另一个式子约束参数,此时我们就需要构建一个新的式子来利用所给的条件,然后将新式子转换为旧式子。 @@ -196,7 +204,37 @@ $\therefore a=1,b=e$。 \subsection{拉格朗日中值定理} -证明不等式最重要的还是找到$f(x)$,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换。 +证明不等式最重要的还是找到$f(x)$,即出现差值$f(a)-f(b)$,那么$f(x)$就是我们的目标函数,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换。 + +\subsubsection{式子转换} + +使用初等运算将目标式子转换减式。 + +\textbf{例题:}设$f(x)$在闭区间$[0,c]$上连续,其导数$f'(x)$在开区间$(0,c)$内存在且单调减少,又$f(0)=0$,证明$f(a+b)\leqslant f(a)+f(b)$,$0\leqslant a\leqslant b\leqslant a+b\leqslant c$。 + +解:不存在两端点相等的条件,所以使用拉格朗日中值定理。 + +因为所要证明的式子中含有$a$、$b$、$a+b$,$f(0)=0$,所以对这几个区间进行拉格朗日中值定理。 + +证明式子中没有减的形式只有和的形式,所以需要对其转换。 + +$f(a)-f(0)=f'(\xi_1)(a-0)$,$f(a+b)-f(b)=f'(\xi_2)(a+b-b)$。 + +从而$f(a)=f'(\xi_1)a$,$f(a+b)-f(b)=f'(\xi_2)a$。 + +又$f'(x)$单调减少,所以$f'(\xi_1)>f'(\xi_2)$。 + +$f(a)\geqslant f(a+b)-f(b)$,所以$f(a+b)\leqslant f(a)+f(b)$。 + +\subsubsection{求原函数} + +这种题目就是证明某个式子成立,式子一边是常数一边是导数式子,要证明,就要将导数式子转换为原函数,方法跟罗尔定理使用的转换原函数的技巧一样。 + +\textbf{例题:}设$f(x)$在$[0,1]$上连续且可导,证明存在一点$\varepsilon\in(0,1)$,使得$f(1)=3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$。 + +证明:由$3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$,可推出原函数为$x^3f(x)$,令$F(x)=x^3f(x)$,则其在$(0,1)$也可导。 + +即使用拉格朗日中值定理,$F(1)-F(0)=F'(\varepsilon)$,$\varepsilon\in(0,1)$。即$f(1)=3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$。 \subsubsection{对数函数特性} @@ -214,12 +252,26 @@ $\therefore a=1,b=e$。 所以$\dfrac{a-b}{a}<\dfrac{a-b}{\xi}<\dfrac{a-b}{b}$,从而$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$,得证。 +\subsubsection{划分区间} + +证明存在两个不同的点在同一个区间满足一个不等式。如果两个点彼此存在一定关系,如上面式子转换的例子$a+b$,$a$,$b$,那么我们可以使用转换,如果两个完全独立的变量,则这种方式没用,我们可以考虑划分区间,假定这两个点在不同的区间,中间以一个区间变量分隔,由于拉格朗日中值定理中两个变量只会出现一次,而间隔变量会出现多次,所以对其分别拉格朗日中值定理,就可以把两个变量换成以间隔变量表示的形式,将两个无关变量的式子变成一个变量的式子。 + +\textbf{例题:}设函数$f(x)$在$[0,1]$上连续,在$(0,1)$内可导,且$f(0)=0$,$f(1)=1$,证明存在不同的$\varepsilon_1$、$\varepsilon_2$,使得$\dfrac{1}{f'(\varepsilon_1)}+\dfrac{1}{f'(\varepsilon_2)}=2$。 + +证明:使用$\varepsilon$将$[0,1]$划分为$[0,\varepsilon]$和$[\varepsilon,1]$两个区间,假定$\varepsilon_1$、$\varepsilon_2$分别在这两个区间上。 + +分别对其进行拉格朗日:$f(\varepsilon)-f(0)=f'(\varepsilon_1)(\varepsilon-0)$,即$\dfrac{1}{f'(\varepsilon_1)}=\dfrac{\varepsilon}{f(\varepsilon)}$,$f(1)-f(\varepsilon)=f'(\varepsilon_2)(1-\varepsilon)$,即$\dfrac{1}{f'(\varepsilon_2)}=\dfrac{1-\varepsilon}{1-f(\varepsilon)}$。 + +即$\dfrac{1}{f'(\varepsilon_1)}+\dfrac{1}{f'(\varepsilon_2)}=\dfrac{\varepsilon}{f(\varepsilon)}+\dfrac{1-\varepsilon}{1-f(\varepsilon)}$,任取$f(\varepsilon)=\dfrac{1}{2}$,原式等于2,得证。 + \subsubsection{查找特定值} -对于证明一种不等式,如果里面没有差式,也无法转换为差式,那么就可以考虑制造差式,对于$f(x)$一般选择更高阶的,$a$选择$x$,$b$要根据题目和不等式设置一个常数。 +对于证明一种不等式,如果里面没有差式,也无法转换为差式(没有相同的$f(x)$),那么就可以考虑制造差式,对于$f(x)$一般选择更高阶的,$a$选择$x$,$b$要根据题目和不等式设置一个常数。 一般是0或1。可以先尝试1。 +对于这种不等式子看上去一般不会想到拉格朗日中值定理。 + \textbf{例题:}当$x>1$时,证明$e^x>ex$。 证明:题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 diff --git a/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.pdf 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zjef7RSG?z(nIi^C=9L8hnqr%M#kph2D<*CM=Pi`k|Fhn3vZv{>gJ239nX<8&a+$Gl zGBR_RGcmH6GPAL>ahb3%a&WP-u$VFO^Zx&Oq_wbv5CU_gt+IoV0XY7j$T{1RnqsmN zYVVzkx~LK&uo5s5+OLiw(K@t_j*g+;VL{-IUp$J4bv$(vd5>av7Rg4qL%z2|8k-+4 zulHFE-yc_B&sl3-G{nlPhe%NI59v`5j0iUeCceLCZX+aEHIv`CybVMF7)JkA_%vbt zYB}El#!$2o{+iG-g#zG)Sg?cjm?3?bK}V{jv)A~M5Ouujy^)wGgnE&6bA^EAd@DWkADF~cU*OmLN| zBL~!U{gsQoNu3<9Buv6&{BzZ15v8i$U(Lz@7#6iKbj^r97V4xKt+E&`>Ijai#Xj2< zHG!(dp$6)#e*#K;xUqP3D9a<$t6&|*zb?bJ>LAz$nK&`S_Ze+NVw`AW2kuruDRxD( zL*$&8<@?H3VN3Tcv*RLGadmbI;uUiRTPBYYe&u7_PbF<(r;b9)!k3+tZ}UoI2<_@}>C zav)T)!}-@=-h##x$g8s>Tbz~1Shi9Cm4)i`Wu^cZvX}3gv-L!w0*ier|Swr|Gh7nx(5=seQg#k)4fmyr+F(0)=0$。 \subsection{中值定理} -一般使用拉格朗日中值定理或泰勒公式。 - -\textbf{例题:}设$f(x)$在闭区间$[0,c]$上连续,其导数$f'(x)$在开区间$(0,c)$内存在且单调减少,又$f(0)=0$,证明$f(a+b)\leqslant f(a)+f(b)$,$0\leqslant a\leqslant b\leqslant a+b\leqslant c$。 - -因为所要证明的式子中含有$a$、$b$、$a+b$,$f(0)=0$,所以对这几个区间进行拉格朗日中值定理。 - -$f(a)-f(0)=f'(\xi_1)(a-0)$,$f(a+b)-f(b)=f'(\xi_2)(a+b-b)$。 - 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a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex +++ b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex @@ -780,6 +780,44 @@ $\left[\dfrac{x^3}{3}-\dfrac{ax^2}{2}+2bx\right]_0^2=\dfrac{8}{3}-2a+4b=a$,$\l 则$f(x)=x^2-\dfrac{4}{3}x+\dfrac{2}{3}$。 +\subsubsection{中值定理} + +中值定理一般是在微分中使用,积分中也可能考到,但是重点是将定限积分化为两个常数的差的形式,所以基本上使用拉格朗日中值定理。 + +\textbf{例题:}设函数$f(x)$在$[0,3]$上连续,在$(0,3)$内有二阶导数,且$2f(0)=\int_0^2f(x)\,\textrm{d}x=f(2)+f(3)$。 + +证明:(1)存在$\eta\in(0,2)$,使得$f(\eta)=f(0)$;(2)存在$\varepsilon\in(0,3)$,使得$f''(\varepsilon)=0$。 + +证明: + +(1)中值定理基本上是对普通函数的差式进行运算,所以令$F(x)=f_0^xf(t)\,\textrm{d}t$($0\leqslant x\leqslant2$),所以$\int_0^2f(x)\,\textrm{d}x=F(2)-F(0)$。 + +由拉格朗日中值定理,存在$\eta\in(0,2)$,使得$F(2)-F(0)=F'(\eta)(2-0)=2f(\eta)$,即$\int_0^2f(x)\,\textrm{d}x=2f(\eta)$。 + +题目条件可知$f(\eta)=f(0)$。 + +(2)还剩下等式右边的条件没有使用,且数字为2和3。如何使用? + +由函数导数$f''(\varepsilon)$为0很显然知道要使用罗尔定理,但是这是二阶导数,就要求得到$f'(0)=f'(3)$两端相等的条件。 + +请注意这里是存在$\varepsilon\in(0,3)$使得条件成立,则相等的两端不一定就等于0和3,在区间范围$(0,3)$内都成立。 + +往往第二问的条件基于第一问的结论,看到第一问,出现了两端相等且区域$\eta$为$(0,3)$子区间,两端相等所以得到这里有一个值的$f'(?)=0$。 + +此时我们必须找到另一个$f'(?)=0$也成立,此时就能看到最开始剩下的条件$f(2)+f(3)=2f(0)$,这个区域正好是$(0,2)$与$(0,3)$的差集。 + +我们需要根据这个条件知道存在一个点让$f'(?)=0$成立。即根据罗尔定理,让存在一个点使得$f(?)=f(0)$。根据右边的条件可以使用介值定理。 + +按这个思路倒退得到证明过程。 + +由于$f(x)$在$[0,3]$上连续则$[2,3]$上连续,则其必然存在最大值$M$和最小值$m$,使得$m\leqslant f(2)\leqslant M$,$m\leqslant f(3)\leqslant M$,所以$m\leqslant\dfrac{f(2)+f(3)}{2}\leqslant M$。 + +由介值定理可知存在$\tau\in[2,3]$,使得$f(\tau)=\dfrac{f(2)+f(3)}{2}=f(0)$。所以此时找到这个点。 + +由(1)$f(0)=f(\eta)=f(\tau)$,其中$0<\eta<2\leqslant\tau\leqslant3$。 + +根据罗尔定理,必然存在$\varepsilon_1\in(0,\eta)$,$\varepsilon_2\in(\eta,\tau)$,使得$f'(\varepsilon_1)=f'(\varepsilon_2)=0$,再根据罗尔定理存在$\varepsilon\in(\epsilon_1,\epsilon_2)\subset(0,3)$使得$f''(\varepsilon)=0$。 + \subsection{变限积分} \subsubsection{极限}